<PRE><B>Question 229129</B>
Let P = speed of the plane.
Let W = Speed of a full headwind.
Let P = Speed of the plane.
Let R1 = Overall speed going against a full headwind = P-W
Let R2 = Overall speed going against half a full headwind = P-(W/2)
Let T1 = Amount of time it takes against a full headwind.
Let T2 = Ampount of time it takes against half a full headwind.
Let D = distance.


Distance is 2900 miles
With a full headwind it takes 5 hours.
With half a full headwind it takes 4 hours 50 minutes = 4.8333333 hours.


We have:


T1 = 5 hours
D = 2900
T2 = 4.8333333 hours


Rate * Time = Distance


Formula against a full headwind is:


R1 * T1 = D which becomes:


R1 * 5 = 2900


Since R1 = P-W, this formula becomes:


(P-W) * 5 = 2900 (equation 1)


Formula against half a full headwind is:


R2 * T2 = D becomes:


R2 * 4.8333333 = 2900


Since R2 = (P-W/2), this formula becomes:


(P-W/2)*4.8333333 = 2900 (equation 2)


We can solve for (P-W) to get:


(P-W) = 2900 / 5 = 580 miles per hour.


We can solve for (P-W/2) to get:


(P-W/2) = 2900 / 4.8333333 = 600 miles per hour.


Since (P-W) = 580, we can solve for P to get:


P = W + 580 (equation 3)


Since (P-W/2) = 600, we can solve for P to get:


P = W/2 + 600 (equation 4)


Since equation 3 and equation 4 both equal to P, then they both equal to each other and we get:


W + 580 = W/2 + 600


Multiply both sides of this equation by 2 to get:


2W + 1160 = W + 1200


Subtract W from both sides of this equation and subtract 1160 from both sides of this equation to get:


2W - W = 1200 - 1160


Combine like terms and simplify to get:


W = 40


Substitute 40 for W in equation 3 to get:


P = W + 580 (equation 3) becomes:


P = 40 + 580 = 620


So far we have:


P = 620
W = 40


To confirm these answers are correct, we substitute in equations 1 and 2 to get:


(P-W) * 5 = 2900 (equation 1) becomes:
(620-40) * 5 = 2900 which becomes:
580 * 5 = 2900


Divide both sides of this equation by 5 to get:


580 = 2900/5 = 580 which is true so answer is confirmed for equation 1.


(P-W/2)*4.8333333 = 2900 (equation 2) becomes:
(620-20)*4.8333333 = 2900 which becomes:
600*4.8333333 = 2900


Divide both sides of this equation by 4.8333333 to get:


600 = 2900 / 4.8333333 = 600 which is true so answer is confirmed for equation 2.


Answer to the question is:


Airplane speed is 620 miles per hour.
Headwind speed is 40 miles per hour.



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